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Explicit Integration

Advances simulation state using derivatives evaluated at the current time.

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Quick Facts

Compact metadata for implementation and discovery.

Category: Simulation
Difficulty: Beginner
Family: Time Integration
Tier: Core
Status: Verified
Review: Source Checked
Sources: 1
Formulas: 1
Relationships: 3
Code example: Ready
Visualization: Ready

Overview

Explicit integration updates positions or other state variables from current velocities, forces, or derivatives without solving an implicit future-state equation.

Core Concepts

time stepcurrent derivativestate updatestability limit

Quick facts

Complexity

Runtime and memory behavior at a glance.

Best

O(state)

Average

O(state)

Worst

O(state)

Space

O(state)

When to Use

Use when forces are simple and small timesteps keep the simulation stable.
Use when implementation simplicity matters more than stiff-system stability.

When Not to Use

Avoid for stiff systems unless timesteps are very small.
Avoid when long-term energy behavior must be tightly controlled.

What It Does

Advances simulation state directly from derivatives evaluated at the current time
Uses time step and current derivative to guide the work.
Produces updated state.

Use Cases

Physics simulation
Animation
Interactive systems

Inputs

Current state
Forces or constraints
Time step

Outputs

Updated state
Constraint corrections
Collision results

Key Formulas (1)

Explicit Euler update

x_next = x + dt · v

x_next = x + dt * v

Advances state using the current derivative.

Pseudocode

1.Read current state and timestep
2.Evaluate derivatives from the current state
3.Compute next state = current state + timestep * derivative
4.Apply any external bounds or events
5.Return the next state

Canonical Example

Input: position 0, velocity 2, timestep 0.5
Output: new position 1 under constant velocity
Explanation: The update uses current velocity multiplied by the timestep.

Implementation Notes

Keep current state and next state separate during the step.
Document whether the method is Euler, semi-implicit Euler, RK2, or RK4.
Test timestep sensitivity on a simple oscillator.

Common Pitfalls

Increasing timestep can make an otherwise stable simulation explode.
Force evaluation and state update order changes the method being implemented.
Explicit Euler is simple but often too dissipative or unstable for demanding physics.